Primitive Root - Cryptography | Number Theory

How to find Primitive root of a given number in mod(n)
Using tables of indices to solve congruences.
Lecture 2 - To find primitive root of a number ' n' : • Primitive Root | Lectu...
Lecture - To find number of primitive roots of a prime number 'p' : • To find number of Prim...
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Extended Euclidean Algorithm for Modular Inverse : • Extended Euclidean Alg...
s DES - Key Generation: • S-DES Encryption || Si...
s DES - Encryption : • S-DES Encryption || Si... Public
Differences between s-DES and DES: • Differences between s-...
Primitive Root : • Primitive Root - Crypt...
Diffie Hellman Key Exchange : • Network Security - Dif...
Euler's Totient Function [Euler's Phi Function] : • Euler's Totient Functi...
Fermat's Theorem - Explanation with example: • Fermat's Theorem - Exp...
Route Cipher: • Route Cipher - Transpo...
Four Square Cipher: • Four Square Cipher - S...
caesar Cipher: • Cryptography - Caesar...
Meet in the middle attack : • Meet in the Middle att...
Blowfish Algorithm : • BLOWFISH ALGORITHM in ...
Elliptic Curve Cryptography - 1 : • Elliptic Curve Cryptog...
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Double DES and Triple DES : • Double DES and Triple...
Steganography & Cryptography : • Steganography and Cryp...
Block Cipher Modes of Operation : • Block Cipher Modes of ...
Message Authentication using Hash Function: • Message Authentication...
Message Authentication Codes MACs : • Network Security - Mes...
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Categories of Cryptographic Systems : • Categories of Cryptogr...
Data Encryption Standard DES : • Data Encryption Standa...
Public Key Cryptosystems : Advantages: • Public Key Cryptosyste...
Public Key Distribution - Part 1 : • Public Key Distributio...
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Vigenere Cipher: • Vigenere Cipher - Cryp...
Affine Cipher : • Affine Cipher Encrypti...
Chinese Remainder Theorem : • Video
Congruences: • Congruences - Cryptogr...
Transposition Cipher : • Transposition Cipher |...
Euclidean Algorithm for GCD : • Euclidean Algorithm to...
Playfair Cipher : • Cryptography - Playfai...
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Thanks sir. In whole You tube, there in no any video about primitive root with this much clear explanation. You saved me. Thanks again.

[@lecturesbyshreedarshank8977]

Thanks ☺️

[@Zatarra69]

this made so much more sense than the last video I watched on this, great job and thank you!!

[@sreelekhap938]

Thank you
That's a great help!!

[@lecturesbyshreedarshank8977]

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Thank you sir very useful

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[@stevemurimi1815]

Good job!

[@sukumar9495]

tnq....sir

[@janidcp6027]

Why we take less than 7 values only?

[@Zatarra69]

Since 8 is just 7 + 1, it would give you all the same values as 1 since the seven gets divided out. 9 would be the same as 2, 10 would be the same as 3, and 15 would be the same as 1 again.

[@Jorsherr]

Helped with primitive roots. Thanks

[@nimeshranatunge734]

Thanks sir

[@lecturesbyshreedarshank8977]

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[@debjoydhole2767]

you should also explain with a composite number

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Managed to finish my exercise thanks to you, thanks !! awesome technique

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Great explanation, thanks.

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Great explanation. Thanks!!!!

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[@RickyCruz1301]

great tutorial!

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[@chinmaysoni4416]

how can 10 and 5 be co-prime to each other gcd(10,5)=5 not 1

[@niteshhebbare697]

Awesome explanation 👍

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[@farahtasnim23]

sir please tell about 31 primitive root

[@lecturesbyshreedarshank8977]

U mean primitive roots of number 31?

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[@zanti4132]

Just a few observations:
Given an odd prime number n, then by Fermat's Little Theorem, aⁿ⁻¹ = 1 mod n for any positive integer a. Therefore, the n-1 entry in your table must be 1 regardless of the value for a. Since every integer from 1 to n-1 must be represented for a to be a primitive root, it follows that 1 can't appear in any other column.
Again because every integer from 1 to n-1 must be represented, it follows that the entry with the n-1 value must be at the halfway point, i.e. the (n-1)/2 entry, for a to be a primitive root. We see this in your example: You show 3 and 5 to be the primitive roots of 7, and 6 is the value for n = 3 when a = 3 or 5. As for why this is true, it is because (n-1)² = n² - 2n +1 = 1 mod n. Since 1 must be the n-1 entry, the n-1 value must be the (n-1)/2 entry.
So to find the primitive roots when n is prime, test the values for a from 1 to (n-1)/2 in ascending order. If you get a value that gives 1 mod n, you can stop - a is not a primitive root. If you get a value that gives (n-1) mod n, then a is a primitive root if and only if a = (n-1)/2.